Properties of Planes

Understanding Planes in Three-Space

A plane in three-space can be defined by a normal vector and a point on the plane.

Scalar Equation of a Plane

A plane's equation can be written as:

Ax + By + Cz + D = 0

Example

Given normal vector n = [3,2,5] and point P0(1,2,3), the equation is:

3(x-1) + 2(y-2) + 5(z-3) = 0

Intercepts of a Plane

To find the \(x\), \(y\), and \(z\)-intercepts of a plane, set the other coordinates to zero:

• x-intercept: (\(D/A, 0, 0\))
• y-intercept: (\(0, D/B, 0\))
• z-intercept: (\(0, 0, D/C\))

Normal Vector and Parallel Vectors

A Normal Vector is perpendicular to the plane. Two direction vectors within the plane are parallel but not parallel to the normal.

Example

Given the plane \(2x - y + 3z - 6 = 0\), the normal vector is \([2, -1, 3]\).

Finding a Plane from Three Points

To determine the equation of a plane given three points \(A\), \(B\), and \(C\):

• Compute vectors \(AB\) and \(AC\)
• Find the cross product to get the normal vector
• Use one point to find \(D\)

Example

Given points \(A(1,1,1)\), \(B(2,3,4)\), \(C(3,5,6)\), compute the plane equation.

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